Prove that the polynomial $z^{2n} + z^n + 1\ (n \in \mathbb{N})$ is divisible by the polynomial $z^2 + z + 1$ if and only if $n$ is not a multiple of $3$.

Let $f(x)$ be a real-valued function defined for $0<x<1.$ If $$ \lim_{x \to 0} f(x) =0 \;\; \text{and} \;\; f(x) - f \left( \frac{x}{2} \right) =o(x),$$ prove that $f(x) =o(x),$ where we use the O-notation.

In a certain forest there are at least three crossroads, and for any three crossroads of roads A, B and C there is a road from A to B without passing through C. A deer and a hunter are standing at crossroads of different paths. Is it possible that they can exchange positions without their paths crossing at other points, that are not their initial positions?

[b]p1.[/b] Twelve people, three of whom are in the Mafia and one of whom is a police inspector, randomly sit around a circular table. What is the probability that the inspector ends up sitting next to at least one of the Mafia? [b]p2.[/b] Of the positive integers between $1$ and $1000$, inclusive, how many of them contain neither the digit “$4$” nor the digit “$7$”? [b]p3.[/b] You are really bored one day and decide to invent a variation of chess. In your variation, you create a new piece called the “krook,” which, on any given turn, can move either one square up or down, or one square left or right. If you have a krook at the bottom-left corner of the chessboard, how many different ways can the krook reach the top-right corner of the chessboard in exactly $17$ moves? [b]p4.[/b] Let $p$ be a prime number. What is the smallest positive integer that has exactly $p$ different positive integer divisors? Write your answer as a formula in terms of $p$. [b]p5.[/b] You make the square $\{(x, y)| - 5 \le x \le 5, -5 \le y \le 5\}$ into a dartboard as follows: (i) If a player throws a dart and its distance from the origin is less than one unit, then the player gets $10$ points. (ii) If a player throws a dart and its distance from the origin is between one and three units, inclusive, then the player gets awarded a number of points equal to the number of the quadrant that the dart landed on. (The player receives no points for a dart that lands on the coordinate axes in this case.) (iii) If a player throws a dart and its distance from the origin is greater than three units, then the player gets $0$ points. If a person throws three darts and each hits the board randomly (i.e with uniform distribution), what is the expected value of the score that they will receive? [b]p6.[/b] Teddy works at Please Forget Meat, a contemporary vegetarian pizza chain in the city of Gridtown, as a deliveryman. Please Forget Meat (PFM) has two convenient locations, marked with “$X$” and “$Y$ ” on the street map of Gridtown shown below. Teddy, who is currently at $X$, needs to deliver an eggplant pizza to $\nabla$ en route to $Y$ , where he is urgently needed. There is currently construction taking place at $A$, $B$, and $C$, so those three intersections will be completely impassable. How many ways can Teddy get from $X$ to $Y$ while staying on the roads (Traffic tickets are expensive!), not taking paths that are longer than necessary (Gas is expensive!), and that let him pass through $\nabla$ (Losing a job is expensive!)? [img]https://cdn.artofproblemsolving.com/attachments/e/0/d4952e923dc97596ad354ed770e80f979740bc.png[/img] [b]p7.[/b] $x, y$, and $z$ are positive real numbers that satisfy the following three equations: $$x +\frac{1}{y}= 4 \,\,\,\,\, y +\frac{1}{z}= 1\,\,\,\,\, z +\frac{1}{x}=\frac73.$$ Compute $xyz$. [b]p8.[/b] Alan, Ben, and Catherine will all start working at the Duke University Math Department on January $1$st, $2009$. Alan’s work schedule is on a four-day cycle; he starts by working for three days and then takes one day off. Ben’s work schedule is on a seven-day cycle; he starts by working for five days and then takes two days off. Catherine’s work schedule is on a ten-day cycle; she starts by working for seven days and then takes three days off. On how many days in $2009$ will none of the three be working? [b]p9.[/b] $x$ and $y$ are complex numbers such that $x^3 + y^3 = -16$ and $(x + y)^2 = xy$. What is the value of $|x + y|$? [b]p10.[/b] Call a four-digit number “well-meaning” if (1) its second digit is the mean of its first and its third digits and (2) its third digit is the mean of its second and fourth digits. How many well-meaning four-digit numbers are there? (For a four-digit number, its first digit is its thousands [leftmost] digit and its fourth digit is its units [rightmost] digit. Also, four-digit numbers cannot have “$0$” as their first digit.) [b]p11.[/b] Suppose that $\theta$ is a real number such that $\sum^{\infty}{k=2} \sin \left(2^k\theta \right)$ is well-defined and equal to the real number $a$. Compute: $$\sum^{\infty}{k=0} \left(\cot^3 \left(2^k\theta \right)-\cot \left(2^k\theta \right) \right) \sin^4 \left(2^k\theta \right).$$ Write your answer as a formula in terms of $a$. [b]p12.[/b] You have $13$ loaded coins; the probability that they come up as heads are $\cos\left( \frac{0\pi}{24 }\right)$,$ \cos\left( \frac{1\pi}{24 }\right)$, $\cos\left( \frac{2\pi}{24 }\right)$, $...$, $\cos\left( \frac{11\pi}{24 }\right)$ and $\cos\left( \frac{12\pi}{24 }\right)$, respectively. You throw all $13$ of these coins in the air at once. What is the probability that an even number of them come up as heads? [b]p13.[/b] Three married couples sit down on a long bench together in random order. What is the probability that none of the husbands sit next to their respective wives? [b]p14.[/b] What is the smallest positive integer that has at least $25$ different positive divisors? [b]p15.[/b] Let $A_1$ be any three-element set, $A_2 = \{\emptyset\}$, and $A_3 = \emptyset$. For each $i \in \{1, 2, 3\}$, let: (i) $B_i = \{\emptyset,A_i\}$, (ii) $C_i$ be the set of all subsets of $B_i$, (iii) $D_i = B_i \cup C_i$, and (iv) $k_i$ be the number of different elements in $D_i$. Compute $k_1k_2k_3$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Cynthia and Lynnelle are collaborating on a problem set. Over a $24$-hour period, Cynthia and Lynnelle each independently pick a random, contiguous $6$-hour interval to work on the problem set. Compute the probability that Cynthia and Lynnelle work on the problem set during completely disjoint intervals of time.

For each five distinct variables from the set $x_1,..., x_{10}$ there is a single card on which their product is written. Peter and Basil play the following game. At each move, each player chooses a card, starting with Peter. When all cards have been taken, Basil assigns values to the variables as he wants, so that $0 \le x_1 \le ... \le x_10$. Can Basil ensure that the sum of the products on his cards is greater than the sum of the products on Peter's cards? (Ilya Bogdanov)

Consider a complete graph of $2020$ vertices. What is the least number of edges that need to be marked such that each triangle ($3$-vertex subgraph) has an odd number of marked edges?

Let \(f:\{0, 1\}^n \to \{0, 1\} \subseteq \mathbb{R}\) be a function. Call such a function a Boolean function. Let \(\wedge\) denote the component-wise multiplication in \(\{0,1\}^n\). For example, for \(n = 4\), \[(0,0,1,1) \wedge (0,1,0,1) = (0,0,0,1).\] Let \(S = \left\{i_1,i_2,\ldots ,i_k\right\} \subseteq \left\{1,2,\ldots ,n\right\}\). \(f\) is called the oligarchy function over \(S\) if \[f (x) = x_{i_1},x_{i_2},\ldots,x_{i_k}\ \text{ (with the usual multiplication),}\] where \(x_i\) denotes the \(i\)-th component of \(x\). By convention, \(f\) is called the oligarchy function over \(\emptyset\) if \(f\) is constantly 1. (i) Suppose \(f\) is not constantly zero. Show that \(f\) is an oligarchy function [u]if and only if[/u] \(f\) satisfies \[f(x\wedge y)=f(x)f(y),\ \forall x,y\in\left\{0,1\right\}^n.\] Let \(Y\) be a uniformly distributed random variable over \(\left\{0, 1\right\}^n\). Let \(T\) be an operator that maps Boolean functions to functions \(\left\{0, 1\right\}^n\to\mathbb{R}\), such that \[(Tf)(x)=E_Y(f(x\wedge Y)),\ \forall x\in\left\{0,1\right\}^n\] where \(E_Y()\) denotes the expectation over \(Y\). \(f\) is called an eigenfunction of \(T\) if \(\exists\lambda\in\mathbb{R}\backslash\left\{0\right\}\) such that \[(Tf)(x)=\lambda f(x),\ \forall x\in\left\{0,1\right\}^n\] (ii) Prove that \(f\) is an eigenfunction of \(T\) [u]if and only if[/u] \(f\) is an oligarchy function.

Let $g(t)$ be the minimum value of $f(x)=x2^{-x}$ in $t\leq x\leq t+1$. Evaluate $\int_0^2 g(t)dt$. [i]2010 Kumamoto University entrance exam/Science[/i]

Find the locus of points $P$ in the plane of a square $ABCD$ such that $$\max\{ PA,\ PC\}=\frac12(PB+PD).$$ [i](Anonymous314)[/i]

In a certain country there are two tennis clubs consisting of $1000 $ and $1001$ members respectively. All the members have different playing strength, and the descending order of palying strengths in each club is known. Find a procedure which determines, within $ 11$ games, who is in the $1001$st place among the $ 2001$ players in these clubs. It is assumed that a stronger player always beats a weaker one.

The freshman class of a school consists of $229$ boys and $271$ girls, and is divided into $10$ rooms of $50$ students each, the students in each room are numbered from $1$ to $50$. The physical education teacher wants to select a relay running team consisting of $1$ boy and $3$ girls or $1$ girl and $3$ boys, so that the four students must be two pairs of students with the same number from two rooms. Show that the number of possible teams is odd.

A collection of cardboard circles, each with a diameter of at most $1$, lie on a $5\times 8$ table without overlapping or overhanging the edge of the table. A cardboard circle of diameter $2$ is added to the collection. Prove that this new collection of cardboard circles can be placed on a $7\times 7$ table without overlapping or overhanging the edge.

For function $ f: \mathbb{R} \to \mathbb{R}$ given that $ f(x^2 +x +3) +2 \cdot f(x^2 - 3x + 5) = 6x^2 - 10x +17$, calculate $ f(2009)$.

Let $ABC$ be a triangle with $\angle BAC=120^o$, which the median $AD$ is perpendicular to side $AB$ and intersects the circumscribed circle of triangle $ABC$ at point $E$. Lines $BA$ and $EC$ intersect at $Z$. Prove that a) $ZD \perp BE$ b) $ZD=BC$

Parallelogram ${ABCD}$ is given with ${AC>BD}$, and ${O}$ intersection point of ${AC}$ and ${BD}$. Circle with center at ${O}$and radius ${OA}$ intersects extensions of ${AD}$and ${AB}$at points ${G}$ and ${L}$, respectively. Let ${Z}$ be intersection point of lines ${BD}$and ${GL}$. Prove that $\angle ZCA={{90}^{{}^\circ }}$.

A committee of $3366$ film critics are voting for the Oscars. Every critic voted just an actor and just one actress. After the voting, it was found that for every positive integer $n \in \left \{1, 2, \ldots, 100 \right \}$, there is some actor or some actress who was voted exactly $n$ times. Prove that there are two critics who voted the same actor and the same actress. [i](Cyprus)[/i]

Say that a sequence $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$, $a_7$, $a_8$ is [i]cool[/i] if * the sequence contains each of the integers 1 through 8 exactly once, and * every pair of consecutive terms in the sequence are relatively prime. In other words, $a_1$ and $a_2$ are relatively prime, $a_2$ and $a_3$ are relatively prime, $\ldots$, and $a_7$ and $a_8$ are relatively prime. How many cool sequences are there?

Suppose $\alpha, \beta$ are two positive rational numbers. Assume for some positive integers $m,n$, it is known that $\alpha^{\frac 1n}+\beta^{\frac 1m}$ is a rational number. Prove that each of $\alpha^{\frac 1n}$ and $\beta^{\frac 1m}$ is a rational number.

There is a point inside an equilateral triangle side $d$ whose distance from the vertices is $3, 4, 5$. Find $d$.

Find all real numbers $a, b, c$ that satisfy $$ 2a - b =a^2b, \qquad 2b-c = b^2 c, \qquad 2c-a= c^2 a.$$

Let $D$ be the midpoint of the base $AB$ of the isosceles acute triangle $ABC$. Choose point $E$ on segment $AB$, and let $O$ be the circumcenter of triangle $ACE$. Prove that the line through $D$ perpendicular to $DO$, the line through $E$ perpendicular to $BC$, and the line through $B$ parallel to $AC$ are concurrent.

Find all prime numbers $ p $ such that $ 4p^2 + 1 $ and $ 6p^2 + 1 $ are also prime numbers.

Prove that the numbers from $1$ to $ 15$ cannot be divided into two groups: $A$ of $2$ numbers and $B$ of $13$ numbers such that the sum of the numbers in group $B$ is equal to product of numbers in group $A$.

Let $n \geq 2$ be a positive integer and $a_{1},\ldots , a_{n}$ be positive real numbers such that $a_{1}+...+a_{n}= 1$. Prove that: \[\frac{a_{1}}{1+a_{2}+\cdots +a_{n}}+\cdots +\frac{a_{n}}{1+a_{1}+a_{2}+\cdots +a_{n-1}}\geq \frac{n}{2n-1}\]